[tex] \frac{x}{x-2} + \frac{x-1}{x+1} =1[/tex]
Least common denominator is (x-2)(x+1)
Multiply this times each term.
[tex](x-2)(x+1)( \frac{x}{x-2} )+(x-2)(x+1)( \frac{x-1}{x+1} )=(x-2)(x+1)(-1)[/tex]
After cancelling the like factors from numerator and denominator, you get
[tex](x+1)x+(x-2)(x-1)=(x-2)(x+1)(-1)[/tex]
Simplify the equation:
[tex]x^2+x+x^2-3x+2=-x^2+x+2[/tex]
Add the opposite of all the terms on the right side to each side of the equation:
[tex]3x^2-3x=0[/tex]
Factor out the GCF:
[tex]3x(x-1)=0[/tex]
Set each factor equal to zero and solve:
[tex]3x = 0[/tex] or [tex]x-1=0[/tex]
x = 0 or x = 1
After checking each answer in the original equation you will see both numbers work. It's important to check because sometimes the answers can be extraneous (they don't work).
